arXiv subjects

Bilal Ahmad Rather

Publications and source records attributed to Bilal Ahmad Rather.

At least 19 recordsLinked to original sources

arXiv2026

Block Structure and Spectrum of Zero-Divisor Graphs of Lipschitz Quaternion Rings Modulo $n$

We study the adjacency spectra of zero-divisor graphs associated with Lipschitz quaternion rings modulo $n$. For an odd prime $p$, the identification $\mathbb{L}_p\cong M_2(\mathbb{F}_p)$ gives a kernel--image partition of the nonzero singular matrices and an explicit block representation of the adjacency matrix. This structure leads to a spectral decomposition, closed expressions for the spectral radius, and information on rank, nullity, eigenvalue multiplicities, and graph energy. For powers of $2$, we determine the graph at $n=2$ and use square-zero ideals in $\mathbb{L}_{2^t}$ to construct large complete subgraphs and obtain corresponding spectral and energy bounds. The resulting framework provides a unified structural approach to the odd-prime and two-adic cases and reduces large adjacency-matrix computations to substantially smaller algebraic models.

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arXiv2026

Degree-based weighted adjacency matrices: spectra, integrality, and edge deletion effects

We study degree-based weighted adjacency matrices associated with symmetric edge-weight functions $ϕ(d_u,d_v)$, with emphasis on complete multipartite graphs and spectral changes caused by edge modification. We characterize its families with three distinct eigenvalues and identifies integral matrices. For complete graphs, an exact threshold is obtained that determines whether deleting one edge increases, preserves, or decreases both the spectral radius and the weighted energy; the generalized Randić family is classified completely, thereby correcting and refining earlier published results in [Bilal and Munir, Int. J. Quantum Chem. (2024)]. We further determine the $ISI$ spectrum under edge deletion from regular complete multipartite graphs, derive the complete weighted spectrum, energy, and inertia of crown multipartite graphs, and prove that adding an edge between two leaves of $S_n$ strictly increases its $ISI$ energy.

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arXiv2026

Metric Bases of Barycentric and Matching Subdivisions of Zero-Divisor Graphs

In this paper, we study metric bases and related metric properties for barycentric and partial matching subdivisions of the zero-divisor graph of $\mathbb Z_{pq}$, where $p$ and $q$ are distinct odd primes with $q>p$. We first recall the natural partition of the zero-divisor graph into the two prime classes and then give a detailed characterization of those subsets of $BS(Γ(\mathbb Z_{pq}))$ that form metric bases when $q\geq 2p-1$. The proof is expanded by separating the role of closed neighborhoods, rows of subdivision vertices, and forbidden twin configurations. We then investigate $M$-subdivision graphs obtained by subdividing selected edges of $Γ(\mathbb Z_{pq})$. In addition to the lower bounds for subdivisions of $p-3$ and $p-2$ edges, we prove an exact formula for matching subdivisions of arbitrary size $r$, $0\leq r\leq p-2$, namely $\dim(G_r)=p+q-r-4$. Several consequences are included to illustrate how a small matching subdivision can reduce the localization cost of the original zero-divisor network.

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arXiv2026

Homological invariants of edge ideals of the multiple extended complete split-like graphs

We study the graphs $MECS_{b,n}^a \cong \overline{K}_a \join \big(n(K_b+K_2)\big)$, obtained by attaching an independent set of size $a$ to $n$ disjoint copies of the block $K_b+K_2$. For $n=1$, we get $MECS_{b,1}^a$, and recover the results of one-block case studied in [Anand, Gupta, Rather and Singh, Homological invariants of some complete split-like graphs, Beitr. Algebra Geom. (2025)]. Using Hochster's formula, tensor products of minimal free resolutions over disjoint variable sets, and the Betti-number formula for graph joins, we derive explicit descriptions of the independence complex, independence polynomial and its analytic properties, Hilbert series, linear and quadratic Betti strands, regularity, projective dimension, and several structural invariants of $MECS_{b,n}^a$. We further classify the well-covered and unmixed members, compute induced matching numbers, show that the family is never Cohen--Macaulay, and record algorithmic procedures for evaluating the Betti data.

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arXiv2026

Explicit Formulas and Unimodality Phenomena for General Position Polynomials

The general position problem in graphs seeks the largest set of vertices such that no three vertices lie on a common geodesic. Its counting refinement, the general position polynomial $ψ(G)$, asks for all such possible sets. In this paper, We describe general position sets for several classes of graphs and provide explicit formulas for the general position polynomials of complete multipartite graphs. We specialize to balanced complete multipartite graphs and show that for part size $r\le 4$, the polynomial $ψ(K_{r,\dots,r})$ is log-concave and unimodal for all numbers of parts, while for larger $r$, counterexamples show that these properties fail. Finally, we analyze the corona $G\circ K_1$ and prove that unimodality of $ψ(G)$ is retained for some classes, and counterexamples exists for complete bipartite and complete multipartite graphs. The results verify the analogy between general position polynomials and classical position-type parameters, and establish balanced multipartite graphs and coronas as potential subjects for further investigation.

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arXiv2026

Graded Betti numbers of generalized split--join graphs and applications

We determine the full graded Betti tables of graph families that subsume several classes studied recently in the literature, namely the generalized multiple complete split-like graphs and the generalized clique-star graphs with arbitrary clique block sizes. The method combines Hochster's formula with a precise decomposition of the associated independence complexes into disjoint unions of simplices and iterated joins of discrete complexes. This reduces every graded Betti number to an explicit coefficient extraction formula and yields closed expressions for the linear strand, higher strands, Hilbert series, regularity, projective dimension, and extremal Betti numbers. In particular, we prove a sharp criterion for $2$-linear resolution and identify the regularity corner in terms of the number of nontrivial clique blocks. As applications, we recover and extend earlier results on equal-block split-like graphs, obtain complete formulas for pineapple graphs, and derive consequences for power graphs of cyclic groups, elementary abelian groups, and prime-power dihedral groups.

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arXiv2026

Combinatorial and analytic aspects of independence polynomials of zero divisor graphs

The independence polynomial of a graph encapsulates all independent sets of differing sizes, a task classified as NP-hard in theoretical computer science. This article examines the independence polynomial of zero divisor graphs in commutative rings. We demonstrate that the independent sets, represented as a sequence of coefficients of the independence polynomial, exhibit unimodality and log-concavity. Therefore, for the independence polynomial of some zero divisor graphs, the unimodal conjecture is true. Additionally, the characteristics of the zeros of the independence polynomial are delineated, along with their corresponding annular regions on the plane.

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arXiv2026

The NF-operator and the NF-Numbers of Simplicial Complexes

Let $\bigtriangleup$ be a simplicial complex and let $δ_{\mathcal{NF}}$ denote the NF-operator. The NF-complex $δ_{\mathcal{NF}}(\bigtriangleup)$ is defined as the Stanley--Reisner complex of the facet ideal of $\bigtriangleup$. Iterating $δ_{\mathcal{NF}}$ gives a periodic orbit (up to isomorphism), and the smallest positive integer $t$ for which $δ_{\mathcal{NF}}^{\,t}(\bigtriangleup)\cong \bigtriangleup$ is called the \emph{NF-number} of $\bigtriangleup$ (Habi and Mahmood, Algebra Colloquium, 2022). In this work, we provide various results and determine explicit formulas for the NF-number for several families of graphs. In particular, we compute the NF-number for dumbbell graphs. We also prove that the NF-number of the complete split graph $S_{n,m}$ equals $m+n+2$, and that the NF-number of the double star $D_{p+q}$ equals $p+q+4$. We conclude with remarks, open problems, and conjectures to guide future research.

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arXiv2026

Extremal chromatic bounds for distance Laplacian eigenvalues

For a connected simple graph $G$ on $n$ vertices with chromatic number $χ$, the distance Laplacian matrix is $\DL(G)=\operatorname{diag}(\Tr_G(v_1),\dots,\Tr_G(v_n))-D(G)$, where $D(G)$ is the distance matrix and $\Tr_G(v)=\sum_{u\in V(G)} d_G(u,v)$ is the transmission. The eigenvalues of $\DL(G)$ are ordered as $\partial^{L}_1(G)\ge \partial^{L}_2(G)\ge \cdots \ge \partial^{L}_n(G)=0$. Building on the chromatic lower bound $\partial^{L}_1(G)\ge n+\ceil{n/χ}$ and subsequent developments, we prove a \emph{color-class majorization principle}: if $(\ell_1,\dots,\ell_χ)$ are the color-class sizes in an optimal $χ$-coloring with $\ell_1\ge\cdots\ge\ell_χ$, then the first $\ell_1-1$ distance Laplacian eigenvalues satisfy $\partial^{L}_i(G)\ge n+\ell_1$, for $1\le i\le \ell_1-1$. This gives sharp lower bounds on the number of eigenvalues above the chromatic threshold $b_χ=n+\ceil{n/χ}$, thereby refining distribution theorems of [Aouchiche and Hansen, Filomat, 2017] and [Pirzada and Khan LAA, 2021]. We further refine clique/independent-set based multiplicity results by deriving explicit chromatic criteria in terms of neighborhood compression, and we generalize the extremal problem for minimum $\partial^{L}_1$ at fixed chromatic number by characterizing the balanced complete multipartite minimizers. Finally, we present a Ky Fan type result, and complement-component consequences of the majorization principle.

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arXiv2026

Betti numbers for cochordal zero-divisor graphs of commutative rings

This paper studies the zero-divisor graphs attached to several finite chain-ring families and computes the homological invariants of their edge ideals by using cochordal constructible systems. We begin with a general layered graph $C(q,L)$, whose vertices are arranged according to valuation layers and whose adjacency is governed by the single rule $k+\ell\ge L$, form some integers $k$ and $\ell$. This graph models the zero-divisor structure of a finite chain ring with residue field of order $q$ and nilpotency index $L$. We prove that $C(q,L)$ is cochordal, determine its type sequence, then correct and refine the Betti formula of its edge ideal [Dung and Vu, Cochordal zero divisor graphs and Betti numbers of their edge ideals, Comm. Algebra 54(2) (2026) 736--744]. The results are then specialized to the Gaussian quotient rings $\mathbb Z_{2^m}[i]$ and to the truncated polynomial rings $\mathbb Z_p[x]/(x^c)$. We compute projective dimension, regularity, independence number, height, Hilbert series, and Cohen--Macaulay behavior. The computations show that these quotient rings have $2$-linear resolutions, while Cohen--Macaulayness occurs only in the expected degenerate or complete-graph cases.

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arXiv2026

On roots of domination polynomials for friendship and book graphs

This study examines the domination polynomials of friendship graphs and book graphs, focusing on unanswered questions related to these families [Alikhani, Brown and Jahari, on the domination polynomials of friendship graphs, Filomat \textbf{30}(1) (2016) 169--178]. For the friendship graph $F_n$, with even $n$, we show that the polynomial $D(F_n,x)$ has exactly three real zeros: $0$ and two simple zeros in the intervals $(-2,-1)$ and $(-1,0)$. We further show that these two nonzero zeros have monotonic variation and converge to $-1-\frac{1}{\sqrt2}$ and $-1+\frac{1}{\sqrt2}$, respectively. We obtain the quantitative approximation $(|z|-1)^2\log |z|\le n$ for any complex zeros of $D(F_n,x)$, resulting in the explicit bound $|z|\le 1+\sqrt{\tfrac{n}{\log 2}}$. For book graphs $B_n$, we ascertain the comprehensive limit set of domination roots and establish results about the presence of real roots contingent on parity. We provide a partial answer to the integer-root an issue by establishing that friendship and book graphs have no nonzero integer domination roots, whereas for corona families, the only nonzero integer root is $-2$.

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arXiv2026

Vertex connectivity of the nonzero nonunit core of the comaximal graph of $\mathbb Z_n$

This article settles Problem 7.2 posed by [Banerjee, Special Matrices (2022)] for the induced subgraph $G_2$ of the comaximal graph $Γ(\mathbb Z_n)$ when $n$ is squarefree. Let $n=p_1p_2\cdots p_m$ with distinct primes $p_1<\cdots<p_m$, and let $G_2$ be the graph on the nonzero nonunit residue classes modulo $n$. We use Chinese remainder representation of $\mathbb Z_n$, and encodes each vertex by the set of vanishing coordinates. This converts $G_2$ into a weighted blow-up of a disjointness graph on nonempty proper subsets of $\{1,\dots,m\}$. Within this model, we derive exact class sizes, explicit degree formulas, the minimum-degree layer, and a short-path criterion. The main theorem proves the connectivity of $G_{2}$ as $κ(G_2)=\prod_{i=1}^{m-1}(p_i-1)=\tfrac{ϕ(n)}{p_m-1}$. Consequently, earlier upper bound is sharp, $G_2$ is maximally connected, and its edge connectivity agrees with its minimum degree. We also obtain distance formulas, diameter and radius information, and a linear-time algorithm once the prime factorization is known.

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arXiv2026

On (distance) Laplacian characteristic polynomials of power graphs

The characteristic polynomials of the Laplacian and the distance Laplacian matrices of power graphs of groups of order $ pqr $, where $ p,q $ and $ r $ are { primes,} are obtained. Further, the characteristic polynomials of these matrices for proper power graphs of cyclic and dicyclic groups are given. The important inequalities for the zeros of the distance Laplacian characteristic polynomials of power graphs of finite groups are presented in comments.

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arXiv2026

Independent domination polynomial of comaximal graphs of commutative rings

The comaximal graph $ Γ(R) $ of a commutative ring $R$ is a simple graph with vertex set $ R $ and two distinct vertices $ a $ and $b $ of $ Γ(R) $ are adjacent if and only if $ aR+bR=R $, where $ aR $ is the ideal generated by $ a $ in $ R $. In this article, the independent domination polynomial $ D_{i}(Γ(\mathbb{Z}_{n}),x) $ of $ Γ(\mathbb{Z}_{n}) $ is discussed, along with its unimodal and log-concave properties for certain values of $n$. Some auxiliary results related to $D_{i}(Γ(\mathbb{Z}_{n}),x)$ are presented in terms of their zeros. In addition, we determine the independence polynomial $ I(Γ(\mathbb{Z}_{n}),x ) $ of $ Γ(\mathbb{Z}_{n}) $ for special values of $n$ and provide a general result associated with it. The bounds for the zero of the polynomial $ I(Γ(\mathbb{Z}_{n}),x ) $ are established, and their log-concave and unimodal properties are examined.

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arXiv2026

Spectral Properties of Zero-Divisor Graphs of Truncated Polynomial Rings

Let $R$ be a commutative ring with identity and let $Z^{\ast}(R)$ denote the set of nonzero zero-divisors of $R$. The \emph{zero-divisor graph} $ \varGamma(R)$ is the simple graph with vertex set $V( \varGamma(R))=Z^{\ast}(R)$, where two distinct vertices$x,y\in Z^{\ast}(R)$ are adjacent if and only if $xy=0$ in $R$. In this paper we investigate the zero-divisor graph of the truncated polynomial ring $R=\mathbb{Z}_{p}[x]/\langle x^{c}\rangle,$ for $c\in\mathbb{N}.$ We determine the spectrum of the $A_α$-matrix associated with $ \varGamma(R)$, and, as special cases, explicitly obtain both the adjacency spectrum and the signless Laplacian spectrum of $ \varGamma(R)$. Furthermore, we prove that the Laplacian eigenvalues, as well as the distance eigenvalues, of these graphs are all integers.

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arXiv2026

On Matrices Whose Distinct Eigenvalues Are Fully Captured by Quotient Matrices

Let $M$ be the $n$-square matrix partitioned into $\ell^2$ blocks $b_{ij}$ according to some partition $P=\{C_{1},\dots,C_{\ell}\}$ of index set $\{1,\dots,n\}$. The quotient matrix $Q=(q_{ij})$ is a $k$-square matrix, with $\ell \leq k \leq n-1$, where $(ij)$-th entry is the average row sum (or column sum) of the corresponding block $b_{ij}$ in $M$. The partition $P$ is said to be \emph{equitable} if row sum of each block $b_{ij}$ is constant. In this case, the matrix $Q$ is referred to as the \emph{equitable quotient matrix} of $M$, and the spectrum of $Q$ is the subset of the spectrum of parent matrix $M$. We characterize some classes of matrices such that their equitable quotient matrix $Q$ contains all the distinct eigenvalues of $M$, thereby information can be obtained form the smallest matrix $Q$ without actually analyzing the parent matrix $M.$ We present necessary and the sufficient conditions for distinct eigenvalue of $M$ contained in the spectrum of of $Q$ in terms of eigenspaces. We end up article with some applications, where distinct eigenvalues of a parent matrix can be completely encoded by quotient matrix.

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arXiv2026

Domination polynomial of co-maximal graphs of integer modulo ring

We investigate the domination polynomial of the co-maximal graph $Γ(\mathbb{Z}_n)$ related to the ring of integers modulo $n$. Explicit formulas are derived for $ n = p^{n_1} $ and $ n = p^{n_1}q^{n_2} $, demonstrating that the resulting polynomials exhibit unimodality and log-concavity. For general $n$, we present structural expressions that connect $D(Γ(\mathbb{Z}_n),x)$ to appropriate induced subgraphs. Finally, we examine domination roots and establish bounds for their moduli using the Eneström--Kakeya theorem.

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arXiv2024

Proximity and Remoteness in Graphs: a survey

The proximity $π= π(G)$ of a connected graph $G$ is the minimum, over all vertices, of the average distance from a vertex to all others. Similarly, the maximum is called the remoteness and denoted by $ρ= ρ(G)$. The concepts of proximity and remoteness, first defined in 2006, attracted the attention of several researchers in Graph Theory. Their investigation led to a considerable number of publications. In this paper, we present a survey of the research work.

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